Oscillation of air particles and speed of sound wave A sound wave is essentially air particles oscillating parallel to the direction of travel of the wave.
We learnt that $v = f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.
Suppose the air particles oscillate with a frequency of $f_{particle}$. Is there any relationship between $f_{particle}$ and $v, f, \lambda$ of the entire sound wave? 
I thought about this because intuitively, if the air particles vibrated faster, then the wave should travel faster, but I am unable to come up with any formula that describe this relationship (if it is even true!).
 A: A sound wave is not particles oscillating, is a mechanical oscillation of a medium made of particles. It is important to separate the medium behaviour from the particle behaviour. Medium behaviour results from the averaging over the values of the many particles, and this generates essentially different phenomena.
A way of visualizing it is thinking of a demonstration or a public gathering: although many people are moving through, trying to reach a friend or coming out or in, the bulk behaviour is what matters if you would see from a plane. From above the mass would look static, even if below there is almost nobody standing in one place. The same is with movement, although not every person might move in the direction of the bulk, from above a marching crowd would look so. But the speeds and directions might differ very much.
So when sound travels through a medium, average densities oscillate due to pressure increase and vice versa. But local densities at a microscopic scale might me much larger than the bulk ones, because even if two particles can come very close, many of them cannot be so close together due to the much higher potential energy related.
So your analogy cannot be followed for these reasons, and this is also the cause that we use different formulations to describe groups of 10 or 100 particles, than when we describe media (made of at least ~$10^{23}$).  
